Warwick Lecture
Academic Year 2021/2022


MA4J5-15 Mathematical Structures
of Complex Systems

Overview

Location: Warwick Mathematics Institute

For examination support, look at the end of this page.

This should be one of the world's most sophisticated courses on mathematical modelling, combined with applications on complex systems. The current crisis, the global pandemic, economic transitions and climate change have brought complex systems theory once more into the spotlight, and their mathematical modelling is key to their understanding and possible management.
 
We keep also this year the partition of the lecture into three parts, structural modelling, dynamic modelling and learning/data analysis. All of these parts have proven to be necessary for any complex systems modelling, sich as models in the Life Sciences, in the Social Sciences, in Economy & Finance or Ecology and Infectious Diseases.

In this lecture will learn how to start the modelling process by thinking about the model's static structure, which then in a dynamic model gives rise to the choice of variables. Finally, with the dive into mathematical learning theories, the students will understand that a mathematical model is never finished, but needs recursive learning steps to improve its parametrisation and even structure.

A very important aspect of the lecture is the smooth transition from static to dynamic stochastic models with the help of rule-based system descriptions which have evolved from the modelling of chemical reactions.

Administration

The course is administratively managed via the University of Warwick Moodle pages. Students get basic notifications via Warwick Moodle.

The day to day exchange of teaching material is done via a cloud application, to which course members will be granted access. Login URL:

https://001.uk.sara-computing.com/nextcloud/

The  life teaching part of this lecture is done during Warwick term weeks 1-10, every Thursday 1 p.m. in room B3.02. 

Most of the lecture material is also available online via Youtube at

SARA Systems Streaming


Weekly Overview

Introduction:

Week 1: Mathematical Modelling, Past, Present and Future

• What is Mathematical Modelling?
• Why Complex Systems?.. 
• Philosophy of Science, Empirical Data and Prediction.
• About this course.

Video link to Lecture 1
Video link to Lecture 2


Part I Structural Modelling


Week 2: Relational Structures

• Relational family: hypergraphs, simplicial complexes and hierachical hypergraphs.
• Graph characteristics, examples from real world complex systems (social science, infrastructure, economy, biology, internet).
• Introduction to algebraic and computational graph theory.

Video to Lecture 4

Week 3: Transformations of Relational Models 
• Connections between graphs, hypergraphs, simplicial complexes and hierachical hypergraphs.
• Applications of hierachical hypergraphs.
• Stochastic processes of changing relational model topologies. 


Video to Lecture 6
Structural Modelling with Hierachies


Part II Dynamic Modelling

Week 4: Stochastic Processes
• Basic concepts, Poisson Process.
• Opinion formation: relations and correlations.
• Master eqation type-rule based stochastic collision processes.

Video link to Lecture 8

Week 5: Applications of type-rule based stochastic collision processes
• Chemical reactions and Biochemistry.
• Covid-19 Epidemiology.
• Economics and Sociology, Agent-based modelling.

Video link to Lecture 9
Video link to Lecture 10
Video link to Lecture 16

Week 6: Dynamical Systems (single compartment) 
• Basic concepts, examples.
• Relation between type-rule-based stochastic collision processes in single compartments and ODE
• Applications, connections between dynamical systems and structural modelling (from Part I), the interaction graph, feedback loops. 
• Time scales: evolutionary outlook.

Video link to Lecture 11

Week 7: Spatial processes and Partial Differential Equations:
• Type-rule-based multi-compartment models.
• Reaction-Diffusion Equations.
• Applications.


Part III Data Analysis and Machine Learning

Week 8: Statistics and Mathematical Modelling 
• Statistical Models and Data.
• Classification.
• Parametrisation.

Video Link to Lecture 13

Week 9: Machine Learning and Mathematical Modelling:
• Mathematical Learning Theory.
• Bayesian Networks.
• Bayesian Model Selection.

Week 10: Neural Networks and Deep Learning:
• Basic concepts.
• Neural Networks and Machine Learning.
• Discussion and outlook.

Examination Support:

Video Link

Lecture Partition - Topics

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Part 1
Structural Modelling

In this part we introduce modelling with structural or relational models, such as graphs.
Learn More
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Part 2
Dynamic Modelling

Most real-world systems have dynamic aspects, i.e. they change their state with time. Dynamic descriptions can be either stochastic or deterministic.
Learn More
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Part 3
Data Analysis and Machine Learning

In emprical science we need to test our models with data. The data themselves have usually first to be analysed with statistical techniques. A modern technique for data analysis is machine learning (ML)
Learn More